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Thread: Another nested material implication question

  1. #1
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    Another nested material implication question

    I am interested in expressions of the form
    ((A → B) → C)

    Specifically, given A, I would like to infer C.

    I have found that
    (((A → B) A) → (((A → B) → C) → C))

    is a valid argument. But maybe I am headed off in the wrong direction. What I am trying to get to is a consequent of just C. Can I get there from here? How?

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  2. #2
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    Re: Another nested material implication question

    Quote Originally Posted by anody View Post
    I am interested in expressions of the for ((A → B) → C)
    Specifically, given A, I would like to infer C.
    It is not clear as to exactly what you mean, Is it this?
    $\displaystyle \begin{array}{*{20}{c}} {(A \to B) \to C}\\ A\\ \hline {\therefore \quad C} \end{array}$ is a valid argument form.
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  3. #3
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    Re: Another nested material implication question

    Yes, except when I evaluate this in a truth table,

    ((((A → B) → C) ∧ A) → C)

    it is not a tautology, hence not a valid argument.

    A B C ((((A → B) → C) ∧ A) → C)
    F F F T
    F F T T
    F T F T
    F T T T
    T F F F
    T F T T
    T T F T
    T T T T
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  4. #4
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    Re: Another nested material implication question

    Quote Originally Posted by anody View Post
    Yes, except when I evaluate this in a truth table,

    ((((A → B) → C) ∧ A) → C)

    it is not a tautology, hence not a valid argument.

    A B C ((((A → B) → C) ∧ A) → C)
    F F F T
    F F T T
    F T F T
    F T T T
    T F F F
    T F T T
    T T F T
    T T T T
    You may think this is not valid BUT it is!
    $\displaystyle \begin{array}{*{20}{c}} {(A \to B) \to C}\\ \neg A\\ \hline {\therefore \quad C} \end{array}$ SEE HERE


    HERE is the proof:
    $\displaystyle \begin{array}{*{20}{l}} {(A \to B) \to C}&\_&{\text{given}}\\ {\neg A}&\_&{\text{given}}\\ {\neg A \vee B}&\_&{\text{addition}}\\ {(A \to B)}&\_&{\text{Material Iplication}}\\ \hline {\therefore \;\quad C}&\_&{\text{Modus Ponens}} \end{array}$
    Last edited by Plato; Oct 25th 2018 at 05:22 PM.
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  5. #5
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    Re: Another nested material implication question

    Wow, beautiful! And no wonder I was unable to get from A to C.

    ...An afterthought:

    ((((A → B) → C) ∧ (A → (A → B))) → C)
    Last edited by anody; Oct 25th 2018 at 07:36 PM.
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